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Discrete spectrum of cranked quantum and elastic waveguides

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Abstract

The spectrum of quantum and elastic waveguides in the form of a cranked strip is studied. In the Dirichlet spectral problem for the Laplacian (quantum waveguide), in addition to well-known results on the existence of isolated eigenvalues for any angle α at the corner, a priori lower bounds are established for these eigenvalues. It is explained why methods developed in the scalar case are frequently inapplicable to vector problems. For an elastic isotropic waveguide with a clamped boundary, the discrete spectrum is proved to be nonempty only for small or close-to-π angles α. The asymptotics of some eigenvalues are constructed. Elastic waveguides of other shapes are discussed.

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Authors and Affiliations

  1. St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg, 198504, Russia

    S. A. Nazarov

  2. St. Petersburg State Polytechnical University, Politekhnicheskaya ul. 29, St. Petersburg, 195251, Russia

    S. A. Nazarov

  3. Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoi pr. 61, V.O., St. Petersburg, 199178, Russia

    S. A. Nazarov

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Original Russian Text © S.A. Nazarov, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 5, pp. 879–895.

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Nazarov, S.A. Discrete spectrum of cranked quantum and elastic waveguides. Comput. Math. and Math. Phys. 56, 864–880 (2016). https://doi.org/10.1134/S0965542516050171

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  • DOI: https://doi.org/10.1134/S0965542516050171

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